A S S O C I A T I O N
O F
Vol.6 Issue 1
M A T H E M A T I C S
E D U C A T O R S
Website: http://math.nie.edu.sg/ame/
MITA (P) No. 153/08/2004
October 2004
President’s Message...
Dear AME Members,
The Association of Mathematics
Educators celebrated its 10th Anniversary
on 29 th May 2004. Under the able
leadership of the past 5 executive
committees the Association has gained
not only national but also international
recognition. Our publications, Maths
Buzz and The Mathematics Educator
have both served their purposes well.
The Maths Buzz has helped to wet the appetite of our members
– kept them posted briefly about upcoming events and shared
with them some interesting snippets on mathematical ideas or
teaching episodes. The Mathematics Educator has been very
successful in publishing research papers related to the teaching
and learning of mathematics. It is currently a much sought after
publication by international researchers. This is certainly a
significant milestone for the Association. To all our past and
present editors of Maths Buzz and The Mathematics Educator
we say, Thank You and Well Done!
The Association has in the last decade provided numerous
professional development and enrichment activities for members
and mathematics teachers. It has co-organized conferences with
Educational Research Association of Singapore (ERA-AME-AMIC
Conference 1999) and Mathematics & Mathematics Education
AG at NIE (ICMI – EARCOME 2002). The Association has also
contributed towards making Mathematics a fun-filled activity
for pupils. A few thousand pupils have already done
Mathematics Trails produced by the Association. To celebrate
the 10th Anniversary, the Association produced a student journal
book which pupils may use for their mathematics journal entries.
Several hundreds of copies have already been purchased by
mathematics teachers for their pupils.
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We need your support and cooperation for the continued wellbeing of the Association and to scale it to greater heights. We
value your views and feedback. Please feel free to e-mail us
about any matter related to the Association. The Association’s
homepage is http://math.nie.edu.sg/ame/ .
Berinderjeet Kaur
President, AME (2004 – 2006)
Contents
AME President’s Address .................................. 1
(2004-2006).. . 1
M e m b e r s o f t h e E x e c u t i v e C o m m i t t e e (2004-2006)
Opening Textbook Sums for Mathematical
Creativity........................................................ 2
Connecting Mathematics and Science .................5
Square Numbers: Their Bonds with Odd Numbers . . 6
O n T e a c h i n g V e c t o r s a n d R e l a t i v e V e l o c i t y . . . . . . . . ..7
7
Forthcoming Events ......................................... 8
Members of the Executive Committee (2004 - 2006)
1
2
At the 11th annual general meeting of the Association, I was
elected the President of the Association. This is my second time
as President of the Association. The sixth executive committee of
the Association will strive to do their best they can for the
Association. A five-day work week coupled with exciting initiatives
such as Teach Less, Learn More and Modes of Alternative
Assessment are currently the challenges the Association is
addressing. TIMSS (Trends in Mathematics and Science Study) 2003
results will be released in December 2004. This time round our
Primary Four and Secondary Two pupils participated in the Study
and we are looking forward to the results.
1
Kaur Berinderjeet
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PRESIDENT
10
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Low-Ee Huei Wuan
COMMITTEE MEMBER
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VICE PRESIDENT
3
Ee Leong Choo, Jessie
Tay Eng Guan
8
Lee Ngan Hoe
TREASURER
Teo Soh Wah
COMMITTEE MEMBER
9
MEMBERSHIP SECRETARY
5
Lim-Teo Suat Koh
COMMITTEE MEMBER
SECRETARY
4
Ng Luan Eng
Chai Yew Loon
COMMITTEE MEMBER
10
Dindyal Jagthsing
COMMITTEE MEMBER
1
Opening Textbook Sums for Mathematical Creativity
Foong Pui Yee
National Institute of Education, Singapore
E-mail: [email protected]
Introduction
Teachers in the primary schools are very conversant with the
standard type of problem sums that are set for pupils to work
individually, after teaching a topic and equipping them with
the necessary basic concepts and procedures to apply. Very often
these tasks are termed as one-step, two-step or multiple-step
problems where pupils are taught certain procedures. For more
complex questions that require analysis of part-whole or
proportional relationships, model drawing is often taught as
the only strategy to solve such type of problems. Examples of
such challenging questions are shown in Figure 1.
3/5 of Pr 6A and 3/4 of P 6B are girls. Both
classes have the same number of girls and P 6A
has 8 more boys than P 6B. How many pupils are
there in P 6A?
defined body of knowledge. This portrayal of school
mathematics has led to lessons where students tediously learn a
collection of techniques by following predetermined rules.
Hence it might be difficult to understand or even attempt to
define what mathematical creativity is about for the learners in
this context. Students should not only know the concepts and
procedures but also how mathematics is created and used to
learn new concepts and to solve problems in original and as
well in as many ways possible. For mathematical creativity to
develop, students must be confronted with rich tasks or
problems. Carpenter and Lehrer (1999) propose that classrooms
need to provide students with tasks that can activate in them
the following interrelated mental activity for understanding and
creative thinking to emerge: (a) develop appropriate
relationships and seeking patterns, (b) extend and apply their
mathematical knowledge, (c) reflect about their own
experiences, (d) articulate what they know, and (e) make
mathematical knowledge their own. The teaching methods for
these thinking activities would include active participation of
students on a variety of strategies and paying attention to the
social aspect of learning. The creative teacher in such classroom
moves away from the image as an expert who knows all and
instead spurs the students to see themselves as autonomous
learners who can think critically and make decisions.
Figure 1: Examples of Challenge Sums
Opening Textbook Sums for Mathematical Creativity
There is however an over reliance on such type of challenge
problems in the Singapore syllabus to assess so called higher–
order analytical thinking skills of the pupils. This also led to an
undesirable practice of over emphasizing a particular method like
model drawing for children to solve similar structured word
problems across related arithmetic topics like whole numbers,
fractions, ratio and percent. In recent years, with the emerging
trends towards emphasis of process and thinking skills in
mathematics, teachers have been encouraged to teach problem
solving heuristics using non-routine problems that would require
strategies such as guess and check; look for patterns; make
supposition; work backwards etc. etc. Very often these nonroutine mathematical problems are used in supplementary lessons
as enrichment for the better ability pupils where the emphasis is
on the practice of heuristics on similar kind of questions.
The author has conducted workshops on the use of open-ended
tasks for many primary teachers in Singapore. For the purpose
of this paper, we will focus on short open-ended questions that
teacher can transform from standard questions commonly found
in textbook exercises. Such questions can be integrated with
normal lessons and do not require an extensive length of several
periods or weeks for pupils to do. Teachers can use short openended problems for its role in developing deeper understanding
of mathematical ideas, generating creative thinking and
communication in students. Caroll (1999) found that short openended questions provided teachers with quick checks into
students’ thinking and conceptual understanding. They were
no more time-consuming to correct than the worksheet
questions that teachers normally give. When used regularly, the
pupils in the study developed the skills of reasoning and
communication in words, diagrams, or picture.
Currently, most mathematics classrooms in Singapore are
practicing the traditional approach of whole class expository
teaching followed by pupil practice of routine exercises and
regular written tests consisting of multiple-choice questions,
short-answer and long-answer open response questions (Chang,
Kaur, Koay and Lee, 2001). There is a need to equip teachers
with a greater variety of mathematical tasks for problem solving
that can enhance their teaching methods. At the same time
students must encounter rich problems where they can offer
evidence of their reasoning and find connections across topics
to develop their mathematical creativity.
Mathematical Creativity
Creativity is not usually associated with the traditional image of
school mathematics which is often seen as a static and well-
2
At the workshops, teachers were given opportunity to create
open-ended problems and then they had to trial their problems
on their own pupils in school. The characteristic features of an
open-ended question are that there are many possible answers:
it can be solved in different ways and if possible, accessible to
pupils in mixed abilities classrooms. It should offer pupils room
for own decision making and natural mathematical way of
thinking. There are three categories of open-ended questions
that teachers can create from textbook sums:
•
•
•
Problem with missing data or hidden assumptions
Problem to explain a concept, procedure or error
Problem Posing
Teachers can create question with missing data or hidden
assumptions from a textbook sum. Consider a routine sum that
one may find in a primary textbook, figure 2. Formulated in such
a closed structure, the teacher and pupils would have in mind
the expected standard response of seeing it as a multiplication
sum. However, in the same context of a Polar bear, teachers in
the Netherlands (Van den Heuvel-Panhuizen, 1996) posed an
open-ended situation for the pupils to solve, see Figure 3.
can transform the question to give pupils the known average
so they would have to find the unknown elements that can have
many answers as shown in this question:
“You have five different brands of the same size bottles of
mineral water. The average cost for a bottle is $1.55. What might
be the actual prices of each of the five bottles of water?”
Creating open-ended questions that ask students to explain a
concept or error can be easily adapted from textbook sums.
Instead of asking for the answer to the textbook questions add
for example “Explain to a friend who does not understand how
you arrived at the answer, you may using diagrams or
manipulatives to help you.” Or a wrong solution can be posed to
pupils that requires them to analyse the errors and then explain
how the problem be solved correctly. Below is an example:
Textbook sum: Round off 5.37 to the nearest tenth.
Convert to open-ended question by adding this statement to the sum:
Maria is helping her brother to do this sum. He wants to know
why you change the 3 to 4 and drop the 7. How can Maria explain
this to her brother?
Figure 2: Closed questions with an expected standard response.
The third category of open-ended questions is problem posing.
This kind of task is gaining popularity in many schools where
students are given opportunities to construct questions based
mathematical concepts or skills that they have learned. For the
younger students they can write number stories when given a
number statement like “5 + 6 = 11”. For teachers to create
problem posing questions they can take a textbook sum, leave
out the question and then ask students to pose whatever
question which they think would fit the given problem situation.
For example:
The figure is made up of a square and an equilateral triangle of
side 5 cm.
Figure 3: Open-ended situation with a missing information
According to Van den Heuvel-Panhuizen, the Polar Bear problem
(Figure 3) represents an important goal of mathematics
education. In an open-ended situation, pupils in addition to
applying calculation procedures are also required to solve
realistic problems where there is no known solution beforehand
and not all data is given. It would require pupils’ own
contributions, such as making assumptions on the missing data.
Without giving the children’s weight, it becomes a real problem
and the pupils have to think about and estimate the weight of
an average child. There is no cue word for students to figure
which operation to use as in the closed question, Figure 2.
A routine textbook sum is normally structured with all the
necessary elements for pupils to find an unknown. Using this
sum, a teacher can leave out some of the given conditions
whereby students have to make realistic assumptions and
decisions that would require their reasoning in order to find
solutions as in the above Polar bear example. Another example,
at the upper primary level for the topic on “average”, the
standard sums are usually set with a given quantity of measures
for pupils to find the average. In an open-ended situation, they
What is the question?.........................................................................
Solve it:
Or the teacher can pose a mathematical statement and ask
students to make up a word problem, for example:
Write a number story for this sum: 1/3 ÷ 3 = ?
Teachers’ Sharing and their Students’ Work
In this section, we share the teachers’ experience in using openended questions with their students. At the workshops, the
teachers found little difficulty in adapting the questions but
initially many of them were apprehensive about giving their
students such tasks. It has never been part of their teaching
where they require students to give explanations and reasons
for their solutions. The practice has always been students are
given problems that have only one answer and one taught
method of finding it. When the teachers had trialed the openended questions with their students, many of them were
surprised by the rich responses that most of their students could
give. Of course, there were also reports by some teachers that
many students seem to lack reasoning and communication skills
initially. The following are experiences of three teachers who
had used open-ended questions and their students’ responses:
3
The Bowls of Oranges
Miss A, a grade four teacher adapted this open-ended question,
figure 4 as an extension to the routine problem sums that she
often gave her pupils. The open-ended version demands higher
order thinking in pupils.
accompanied by mathematical sentences using appropriate
symbols. They were also systematic in their listing of possible
answers. The teacher observed that among the above average
pupils some were quick to catch onto the idea of listing the
factors of 12 to get 6 combinations. This was related to a new
topic under “factors and multiples” that they had just learned
at grade 4. One pupil exclaimed during discussion “this made
understanding factors so easy!”
The Rectangle Problem
The rectangle problem below was used by Mrs. B with her grade
six class.
A rectangular piece of paper is cut into four pieces, A, B, C
and D as shown in the figure. The piece A is given to Alice
and the piece B is given to Bessie. Do you think Alice has a
bigger piece of paper than Bessie? Explain your answer in
at least two ways.
A
D
B
C
The task was open-ended to allow pupils to apply non-algorithm
thinking; to access relevant knowledge on properties of
rectangles and triangles; to apply geometric concepts and
visualization skills to manipulate or dissect a shape; they might
make assumptions about the missing data which they think are
necessary and to display their creativity in using a variety of
strategies for the solution. The pupils had prerequisite
knowledge of properties of rectangle and triangle and they had
learnt the area formula for both shapes. However this is the
first time that pupils were exposed to such open-ended problem.
Figure 4: Open-ended task with higher cognitive demands
A standard closed textbook question opposite to this openended task would normally be of this format:
There are 12 oranges to put into 3 bowls. Each bowl must have the
same numbers of oranges. How many oranges are there in each bowl?
Miss A observed that the pupils were extremely excited over
the open-ended question when told that they were to work in
small groups as Math Investigators to solve this problem. They
wanted to do better than other groups in finding more solutions.
The pupils were thrilled by the idea of being Math Investigators
as reflected in their eagerness to start the project. They liked
the investigative part and the freedom to decide on how to
present their solutions. Initially some pupils had difficulty
understanding the statement in the problem: “each bowl must
have the same number of oranges” and needed more
clarification from the teacher.
To her surprise, most of her pupils were able to respond
appropriately and the variety of the children’s thinking was
enlightening. Some children could reflect upon analysing the
question, that it cannot be solved straightaway as there was
not enough given information. Many groups were able to give
multiple representations and made connections between
repeated addition, multiple and division. They drew pictures
4
When the discussion started, they seemed a little handicapped
as it was also the first time they did group activity for math.
Most pupils seemed confident in handling the preliminary
activity which is a closed problem but when they received this
open-ended version they were hesitant and were doubtful that
they could perform the task. Their most common query was
whether there was missing information that the teacher forgot
to give in the worksheet. They questioned whether they can
solve the problem without given data for the length and
breadth. The pupils felt insecure about handling a mathematical
problem without any numbers involved. It was only after much
assurance that they were convinced that the problem was
solvable. When they were reminded of the preliminary activity,
most groups started using the cutting method but some had
difficulty orientating the pieces. Surprisingly those pupils who
were normally not proficient in math actually got the correct
answer first but they had problem verbalising their methods.
Teacher pressed for justifications no matter what answer the
pupils decided. With the teacher’s guidance and hints, listed
below were four good methods extacted from the solutions of
the pupils:
•
•
•
•
Cutting and matching A to B
Cutting into 8 equal triangles
By formula, assuming some dimensions
Drawing unit squares.
There were two groups of pupils who intuitively perceived that
they are the same by observation. There was also a group who
use abstract algebraic reasoning to justify their conclusion.
Teachers’ Reflection
This approach of opening the textbook sums for pupils to do
has enabled teachers to see pupil’s thinking rather than the
teacher’s own thinking through closed questions that have
predetermined method and answer. To encourage others to
implement open-ended questions in their teaching, excerpts of
two of the teachers, Miss A and Mrs. B’s reflection of their
experience are expressed in the following:
Miss A: ( Bowls of Oranges)
“As we become more and more exam-oriented, activities are
relegated to the end of the priority need. Yet, in doing so,
we are effectively putting blinders on the pupils such that
they are capable only of answering exam style questions. By
a mere twist of the question, making it open-ended, children
are caught off-guard and are unable to answer the question,
as there is “too little information given”
I enjoyed presenting this activity to the pupils as much as
the pupils enjoyed it. As one child exclaimed during the
discussion I had with the pupils, “this made understanding
of factors so easy”. True indeed, this activity was suited as
an activity to launch the concept of factors. One constraint
is the time factor. In our current system, teachers often find
themselves trying to cope or catch up with the syllabus and
scheme of work. To keep pace, often teachers resort to a
teacher-centred style of learning.
The use of open-ended problems should be used more
frequently in the classroom as it encourages pupils to think
out of the box, to be more curious and be aware of the
possibilities of more than one solution.”
Mrs. B ( Rectangle Problem)
“It was an enriching experience for the first time and the
pupils enjoyed it too. I thought to a certain extent this
activity has given them a certain level of confidence and
also helped them to explore instead of just solving routine
problems.
There was reinforcement of the concept of triangle during
class discussion as they discovered they have used the wrong
dimensions. Some groups after getting one answer stopped
working as they have been doing problems with only one
answer. I guess the teaching of math is different nowadays
and pupils as well as the teacher need to change their
mindset in solving problems
I felt that these open-ended problems could be introduced
to more teachers and even implemented in the Math
textbooks. As much as Math is concerned teachers still need
to teach the fundamental concepts and with these knowledge
pupils should be encouraged to explore amazing discoveries.
At the end of the day, I guess what I really want is to teach
them how to fish and not just to provide them with the fish!”
Conclusion
Due to the limitation of space here, there are several other
interesting teachers’ questions and pupils’ responses that could
not be accommodated in this paper. From the samples that we
have presented here, the teachers who have used such short open-
ended questions in their classrooms found many advantages that
enhance their professionalism. It is an approach that enable them
to teach mathematics that aligns with the national vision of
“Thinking schools, learning nation” where the focus is on thinking
and reasoning that characterize the shift from practicing isolated
skills towards developing rich network of conceptual
understanding. From the evidence presented in the pupils’ work,
the pupils were capable of communication in mathematics using
words, diagrams, pictures or manipulatives. Pupils in presenting
their solutions to others could compare and examine each other’s
method and discoveries from thence they could modify and
further develop their own ideas in creative ways.
Note: The author wishes to thank all the teachers who participated in
the in-service programme and especially those teachers whose feedback
and pupils’ work are shown in this paper.
References
Chang, S.C. , Kaur, B., Koay, P.L., & Lee, N.H. (2001). An exploratory
analysis of current pedagogical practices in primary
mathematics classroom. The NIE Researcher, 1(2),7-8.
Carpenter, T.P., & Lehrer, R. (1999). Teaching and learning
mathematics with understanding. In E. Fennema & T.A.
Romberg (Eds.), Mathematics classrooms that promote
understanding (pp.19-33). Mahwah, NJ: LEA Publishers.
Carroll, W.M. (1999). Using short questions to develop and assess
reasoning. In L.V. Stiff & F.R. Curcio (Eds.), Developing
mathematical reasoning in grades K-12, 1999 Yearbook (pp.
247-255). Reston, Va.: NCTM.
Van den Heuvel-Panhuizen, M. (1996). Assessment and realistic
mathematics education. Utrecht: CD-B Press/Freudenthal
Institute, Utrecht University.
Connecting Mathematics
and Science
By Low-Ee Huei Wuan, SP
On 17 August 2004, a group of 30
Mathematics teachers and lecturers
gathered at Singapore Polytechnic (SP) for a
workshop jointly organised by AME and SP, on
making connections between the major subjects
in our curriculum, Mathematics and Science.
The speaker, Mr Russell Brown, is an Australian
Mathematics and Science teacher from Bendigo
Senior Secondary College, Victoria. He shared with
the participants how they could engage their students
by making learning relevant, practical and fun in the classroom.
The activities as Mr Brown had shown, required only minimal
preparation and expertise on the part of the teacher. Resources
used were not sophisticated but basic household items such as like
nail varnish remover, vinegar, baking powder and methylated spirit.
Mr Brown also demonstrated to the audience, the sequence from
the collection of data with a datalogger to the plotting of graphs
with a graphing calculator. In this way, teachers are able to relate
real life data to graphs, and subsequently lead students to form
appropriate mathematical models such as differential equations.
Judging from the fun the participants had, matching their
movement to a drawn graph and vice versa, they have certainly
taken away from the workshop more than just mere connections.
5
Square Numbers: Their Bonds with Odd Numbers
Chua Boon Liang and Ng Swee Fong
National Institute of Education, Singapore
Introduction
Students often recognise 1, 4, 9, 16, 25,... as the sequence of square numbers
because the first term 1 is 12, the second term 4 is 22, the third term 9 is 32
and so on. However, not many may be familiar with another description
of these square numbers in terms of a series of consecutive odd numbers.
This article describes an activity which secondary school students may use
to help them explore and establish this relationship. Students manipulate
with abstract objects of 12, 22, 32, 42, 52,... to get the corresponding sequence
of square numbers – 1, 4, 9, 16, 25,... In this activity however, students
manipulate concrete objects – square tiles – and through manipulating
such concrete objects, students get a sense of the pattern underpinning
the structures they have constructed. Once students get a sense of the
pattern underpinning these structures, they can begin to articulate this
pattern verbally and then to generalise and record this pattern symbolically.
This activity offers students the opportunity to link numerical
representations of numbers with their corresponding geometric structures.
Teaching Points
According to Mason (1996), the primary aim of teaching mathematics is to
expose students to a fundamental process of mathematical thinking generalising. To achieve this aim, he proposes a helical model where
students are given the opportunities to encounter and explore an idea.
Through manipulation of the objects embedded in the idea, students begin
to get a sense of the pattern underpinning this idea and are able to
generalise the pattern. The phase of encountering an idea and exploring
it is known as manipulation. The next phase of getting a sense of the
pattern occurs when a pattern is observed from manipulating the idea.
Lastly, the phase when the pattern is represented verbally, diagrammatically
or symbolically is known as articulation.
Based on Mason’s model, the activity described in this article is designed
to allow students to manipulate objects to generate a sequence of
growing squares, to describe how they form the growing squares and
then to generalise the patterns they see. The aims are manifold:
To get students to
(a) focus on the manipulation of objects,
(b) use language to describe what their
procedures are, and
(c) record the rules symbolically.
Also by doing so, students will link number patterns to their geometric form.
From the algebraic aspect, this activity will additionally help to deepen students’
understanding of the use of letters to represent numbers as well as structural
equivalence. By structural equivalence, it is hoped that, for instance, 32 will be
perceived as 32 =1+3+5 rather than just 32=9.
The Activity
With teachers’ guidance, students will work through a series of tasks to
answer the question – Find the sum of the first n consecutive odd numbers.
Given the following diagrams of three squares, students will perform
the tasks given in the table below.
Figure 1
Figure 2
Figure 3
Manipulating and 1. Use square tiles to make these figures.
getting a sense of 2. Describe how you would make Figure 2 from Figure
1, Figure 3 from Figure 2, Figure 4 from Figure 3.
the pattern
Articulating the 3. Without using the square tiles, describe how
you would make Figure 10.
pattern
Generalising the 4. Find the sum of the first n consecutive odd
numbers.
pattern
6
If different coloured square tiles were to be used, students may be able
to see another pattern and to generate relationships that were not
obvious. For example, the following figures could be formed using tiles
of two different colours.
One way to describe the growing pattern is
Number of tiles in Figure 1 = 12
Number of tiles in Figure 2 = 22 = 1+3
Number of tiles in Figure 3 = 32 = 1+3+5
(equation 1)
(equation 2)
(equation 3)
It is important for students to describe what they see in Figure 3 as a series
of growing consecutive odd numbers (e.g., 1+3+5 ) and not as an outcome
(e.g., 9 or 4+5). This mode of description should continue for Figures 4, 10
and so on. Additionally, teachers can encourage students to describe the
structural equivalence of equation 3. From Figure 3, students may be able
to recognise that the sum of the first three consecutive odd numbers is 32 –
the square of the figure number. To consolidate the pattern they see,
teachers can encourage students to test the pattern with Figures 4 and 10.
Further consolidation may take place if students are asked to identify the
figure number given the series of consecutive odd numbers such as
1+3+5+...+23. In this case, Figure 12 is the answer. With this, a link could
be made between the figure number and the number of consecutive odd
numbers in the series as well as between the first and last terms of the
series. For example, the number of consecutive odd numbers in 1+3+5+...+23
is 12 which is the figure number and this figure number is also equivalent
to the mean of the first and last terms of the series. Hence, given another
figure number, the last term of the series of this figure could be determined.
For example, given Figure 15, the last term of the series should be 2X151=29 and ideally this should be provided by students without having to list
the entire series for Figure 15.
So, what is the sum of the first n consecutive odd numbers? The answer
is clearly n2! However, to find the sum of 1+3+5+...+357 is not so trivial.
In order to answer this question, it is necessary to find out the number of
terms in the series. This is where it is important to identify that the last
term of the first n consecutive odd numbers is 2n-1. In this case, 2n-1 is
equivalent to 357. With this in place, the value of n can be found and
hence, the sum of 1+3+5+...+357 can also be determined.
Conclusion
Rather than describing 1, 4, 9, 16, 25,... simply as a sequence of square
numbers, this article provides another interpretation of these square
numbers from both the geometric and algebraic aspects with an
exploratory activity. By using tiles to form squares, it is possible to show
geometrically that every square number can be expressed as the sum of
consecutive odd numbers. So, the term n2 can be expressed algebraically as
the sum of the first n consecutive odd numbers (i.e., n2=1+3+5+...+(2n-1)).
Apart from this relationship and the fact that the last term is 2n-1 ,
students may observe that n2 can also be expressed as the square of
the mean of the first and last terms of the sequence to be summed
2
(i.e., n 2= first term + last term ). As these patterns can be quite
2
abstract to learn, it is therefore hoped that through manipulating and
exploring objects as well as getting a sense of the patterns from it,
students are able to connect better both the geometrical and algebraic
representations of square numbers.
(
)
Reference
Mason, J. (1996). Expressing generalities and roots of algebra. In N.
Bednarz, C. Kieran & L. Lee (Eds), Approaches to algebra – Perspectives
for research and teaching, (pp. 65 – 86), Dordrecht, The Netherlands:
Kluwer Academic Publishers.
On Teaching Vectors and Relative Velocity
Toh Tin-Lam, with contributions from participants of the in-service workshop on Vectors and Relative Velocity
National Institute of Education, Singapore
Relative velocity was introduced in the pure mathematics
component of the GCE “O” Level Additional Mathematics syllabus
in 2001. Previously, this topic was subsumed under the particle
mechanics section in the earlier Additional Mathematics syllabus.
Questions then tended to focus on the mechanical drawing of
vector diagrams and the extensive use of geometry and
trigonometry. On close examination of the recent GCE “O” Level
syllabus, one would find the focus to be rather different now.
With this backdrop, I would like to discuss two aspects in the teaching
of relative velocity : the intuitive understanding of the concept of
relative velocity and the use of ‘i-j’ vector notation in this topic.
On this note, I shall begin by illustrating some numerical
problems contributed by the in-service participants.
Intuitive Understanding of the Concept of Relative
Velocity
Before introducing VAB = VA – VB, the formal definition of the
velocity of A relative to B, it would be helpful to allow students to
appreciate that VAB is in fact how the observer B feels when A is
moving (via a software such as a GSP applet, for example, see [3]).
Since B is unaware that he is also moving, so what he perceived
will be different from the observation made by any stationary
observer A. Of course, if B is at rest (relative to the Earth), it would
reduce to a trivial case of VAB being the actual velocity of A, i.e. VA.
Otherwise, the two quantities are not the same.
Consider the following question (the original source of this
question is a particular Physics textbook, which the course
participant had forgotten).
Example 1.
Rain is coming at an angle of 30º to the vertical
plane at a speed of 4 m/s. A man is walking on
horizontal ground as shown. How fast must
the man walk / run, in the left direction, so
that the rain will just miss hitting his back?
Let M denote the man and R, the rain. In the case of Example 1,
the student solving this problem would realize that in order for
the rain to just miss hitting his back, the rain must appear to
him as if it was falling vertically, that is, VRM must be acting
vertically downwards. Following that, using either geometric
construction or elementary trigonometric computation, the
man’s speed can be found easily. The reader can effortlessly verify
that he must run at a speed of 2m/s1.
Consider another question, which is quoted from a past year
Cambridge Examination paper.
Example 2. (June 99)
The runner P starts her last 400 m lap of a race 14 m behind a
runner Q. Each runner maintains a constant speed during
the last lap and P wins the race with a lead of 1 m over Q.
Given that the speed of P relative to Q during the last lap is
0.25 ms-1, calculate the speed of P.
The runner Q in Example 2 perceived that P was 14 m behind
him and finally completed the race 1 m ahead of him. So the
total distance that Q perceived P had covered was 15 m. Since
the speed of P relative to Q is 0.25 ms-1, the time taken for P to
15
overtake Q and win with a lead of 1 m = 0.25 = 60 seconds.
However, the actual time taken and the “relative” time taken
are the same! Coming back to reality, P has traveled 400 m
(within the time of 60 seconds). Therefore, the speed of P can
be calculated as
400
60
=
6
2
-1
3 ms .
It is important to help students develop an intuitive feeling of
relative velocity. To begin, it may be useful to start with questions
such as in Example 1 and 2 before plunging into problems that
require more sophisticated constructions and geometry.
The Use of ‘i-j’ Vector Notation in Relative Velocity
Some of the participants are worried about the abundant use
of ‘i-j’ notation to describe the position (displacement) and
velocity of particles in vectors in the GCE ‘O’ Level examination
questions. In such questions, students must be able to distinguish
between displacement and velocity vectors. Students need to
know the meaning of relative velocity in which the description
is given in ‘i-j’ notation. An example is appended below:
Example 3.
A ship sails from point A to point B, the position of B relative
to A is (2400i– 600j) km , where i is a distance of 1 km due
east and j denotes a distance of 1 km due north . Due to a
steady current , the journey takes 6 hours . The velocity of
the ship in still water is (120i+80j) km/h . Find the speed and
direction of the current.
(Answer: 284.7 m/s; 085.9o)
In Example 3, students need to : (1) interpret the displacement
and velocities in ‘i-j’ notations; (2) understand the meaning of
the terms velocity in still water, and other related quantities
like the true velocity of the ship etc., and (3) apply the formula
Change in Displacement = (Time Taken) x Velocity,
which is a vector counterpart of the equation : Distance traveled
= Time Taken x Speed.
Another nature of the problem includes determining the
likelihood of two particles moving with constant velocity
intercepting. If they meet, we would be requested to find the
time of interception or otherwise, determine the shortest
distance between them. A few questions contributed by the
course participants are appended below.
Example 4.
Let i and j denote a distance of 1m along the East and the
North respectively. A thief was running off at 2i m s–1 when
he heard a growl from a menacing dog 5 m away, at a bearing
of 150º. Wherever a vector is required as an answer, write it
in the form of (ai + bj).
(i) Write out the position vector of the dog, relative to the
thief’s position.
(ii) Write out the displacement vector of the thief at the
end of the 3rd second after he heard the dog.
(iii)Assuming the dog managed to bite the thief’s leg 3
seconds after the thief heard it growl, find its
displacement vector for these 3 seconds.
(iv) Calculate its speed, given that it ran at a constant speed
in a straight line.
5
(Answers: 1(i) 2(i √ 3j) (ii) 6im
(iii)
1
7i
2
(
5√ 3 j
)m
(iv) 1.86 m s–1)
(Footnotes)
1
Based on the fact that it is a Physics question, I believe the intention of the author is to test the students’ ability in understanding the concept of resolution of vectors. However, for the teaching
of relative velocity, our objective would be to instill in our students a more intuitive concept of relative velocity.
2
While it is stated in the syllabus that problems of closest approach will not be tested, the entire idea illustrated in Example 7 is by the algebraic approach rather than by the relative velocity
method. Hence the question can be linked to calculus or quadratic expressions.
7
Example 5.
A ship A is traveling with velocity 20j km/h. At 12 00 hour, it
50
i
j km. A second
is at the point with position vector 50
√2 √2
ship B is at the origin. If ship B steers at 25 km/h so as to just
intercept ship A, find
(a) the direction in which B must travel
(b) the time when the interception takes place.
0
(Answer: 100.9 ; 13 26h)
In solving problems like Example 4 and 5, students need to be
familiar with the concepts and apply the formula : at any time t
after the movement of an object with constant velocity v, its
position r(t) can be expressed as
r(t) = a + t v
——————— (1)
where a is the position vector of the starting point of the particle.
The rest of the problems can be solved algebraically – even
without knowing any concept of relative velocity!
However, the tediousness (in terms of algebraic computation)
involved in solving the questions such as in Examples 4 and 5
above, can be reduced by using relative velocity :
Two particles A and B moving with constant velocities and
with starting points A0 and B0 respectively will eventually
intercept if and only if VAB is along the direction of the
vector A0B0. The time at which the interception takes place
is given by |VAB| .
Example 6 above requires students to deal with surdic
expressions in a problem involving kinematics / relative velocity.
Example 7 requires students to work with the situation in which
the two particles do not intercept. Hence the time of closest
approach and the shortest distance between the two ships are
required.2
Example 7.
The RSS Courageous, a navy ship is at the point with position
vector 20j km and an oil tanker Texan One is at the point
with position vector (-100i + 20j )km, where i and j denote
a distance of 1 km along the east and north respectively.
RSS Courageous is traveling at a velocity of 30j km/h and
Texan One is traveling at a velocity of (10i – 20j) km/h. Will
the two ships collide? If yes, find the time of collision;
otherwise, find the time at which they are nearest together.
Find this closest distance between the two ships.
(Answer: the ships are closest when t = 1 h and the shortest
distance between them is √9000 km)
To solve Example 7, one way would be to prove that the two
ships do not intercept (by either the algebraic means or the
Relative Velocity approach) and then proceed by finding an
expression for the distance between the two ships for time, t.
To find the shortest distance between the two ships, one needs
to use either calculus to minimize the expression of t or by the
algebraic approach of completing the squares (since the
expression in terms of t is a quadratic expression).
Conclusion
|A0B0|
The above method can be derived algebraically, see [2; Pg 63-64]. The
derivation only involves some fundamental knowledge of vectors.
If the entire question on the motion of two particles is given in ‘i-j’
notation as in Examples 4 and 5, it can be worked out mentally whether
two particles moving with constant velocity will eventually intercept.
We will leave it to the readers to try the above two examples.
Appended below are two more similar examples.
Example 6.
With respect to an origin, a car is at the point with position
vector 11i + ( 10 √2 10 )j and a van is at the point ( 11√2 5)i
+ k(4 √2 3 )j. If the car is traveling at a constant velocity √2 i
+ j and the van is traveling at a constant velocity 2i + √2 j,
find the value of k such that the two vehicles meet, giving your
answer in the form a+b √2 where a, b and c are integers.
We have discussed the two aspects in the teaching of relative
velocity : the intuitive understanding of the concept of relative
velocity and the use of the ‘i-j’ vector notation in this topic. We
have also included some questions constructed by in-service
course participants. For teachers who would like to acquire more
resources on vectors and relative velocity, the references
indicated below may be of help.
References
[1] National Institute of Education, Teaching of Vectors,
Kinematics and Relative Motion, published as Chapter 14 in
The Green Book: Resources and Ideas for Teaching Secondary
Mathematics, National Institute of Education, 2004.
[2] Toh T.L., Vectors and Relative Velocity, McGraw-Hill, 2004.
[3] Toh T.L., On using Geometer’s Sketchpad to teach relative
velocity, Asia-Pacific Forum on Science Learning and Teaching,
Vol 4(2), Article 8, 2003.
In http://www.ied.edu.hk/apfslt/v4_issue2/toh/
c
(Answer:
19+10 √2
23
)
FORTHCOMING EVENTS FOR YOUR ATTENTION!
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•
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